円周上に「量子力学的固有状態~固有波動関数」を乗せるとすると ただの円か、周波数を少しずつ増やしながらサインカーブが円周上に乗ったようなものが出てくる
固有状態は「固有値分解」するとよいので、かなり細かく点をとったサイクルグラフでその様子を確認してみる
上下のオフダイアゴナルに値1が入っていることと、対角線から一番遠い2点にも1が入っている形になることに注意
library(igraph)
##
## Attaching package: 'igraph'
## The following objects are masked from 'package:stats':
##
## decompose, spectrum
## The following object is masked from 'package:base':
##
## union
n <- 100
edge.list <- cbind(1:n,c(2:n,1))
g <- graph.edgelist(edge.list,directed=FALSE)
ad.mat <- as.matrix(get.adjacency(g))
image(ad.mat)
eigen.out <- eigen(ad.mat)
eg.val <- eigen.out[[1]]
plot(eg.val) # コサインカーブになっていることに注意
固有ベクトルは?
V <- eigen.out[[2]]
matplot(V[c(1:n,1),1:10],type="l")
matplot(V[c(1:n,1),11:20],type="l")
円周に現れる固有状態は、正規直交関数であって、それはフーリエ変換の構成関数
じゃあ、球面に対して同じことをすると、「球面調和関数」が現れそうです
やってみる
球面を模して、それなりに細かい均等点配置のメッシュグラフを作って、固有値分解してみる
library(devtools)
install_github("ryamada22/Ronlyryamada")
## Skipping install of 'Ronlyryamada' from a github remote, the SHA1 (9c4d20f1) has not changed since last install.
## Use `force = TRUE` to force installation
library(Ronlyryamada)
library(igraph)
sp.mesh <- my_sphere_tri_mesh(n.psi=40)
xyz <- sp.mesh$xyz
plot3d(sp.mesh$xyz)
segments3d(sp.mesh$xyz[c(t(sp.mesh$edge)),])
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g <- graph.edgelist(sp.mesh[[2]],directed=FALSE)
ad <- get.adjacency(g)
ad. <- as.matrix(ad)
固有値の分布は何か、意味のあるものなのかもしれないけれど、プロットからすぐには思いつかない。
eigenout<- eigen(ad.)
plot(eigenout[[1]])
k <- 1
L <- ceiling(sqrt(k))-1
M <- 0
ttl <- paste("l=",L," m=", M)
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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k <- k+1
newL <- ceiling(sqrt(k))-1
if(newL > L){
L <- newL
M <- -L
}else{
M <- M + 1
}
ttl <- paste("l=",L," m=", M)
#open3d()
new.xyz <- xyz * (eigenout[[2]][,k])/mean(eigenout[[2]][,k])
plot3d(new.xyz,main=ttl,xlab="",ylab="",zlab="",axes=FALSE)
segments3d(new.xyz[c(t(sp.mesh$edge)),])
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